At some point we should note that the Tahiti-Darwin pair was selected because it approximates a dipole, which is the embodiment of a perfect -1 anti-correlation that you mention.
Yet, Darwin and Tahiti were also chosen because they have a long history of almost continuous atmospheric pressure readings, and actually don't represent the best dipole pair in the Pacific. I have seen Port Moresby mentioned as a better candidate than Darwin, while Tahiti is not as good as it can be either.
I am working extra-curricularly with the climatologists and computer scientists at the U of Minnesota (a very interesting mix) and they have some interesting ideas on how to find the ideal dipoles from the geospatial and temporal records:
J. Kawale, S. Liess, A. Kumar, M. Steinbach, A. R. Ganguly, N. F. Samatova, F. H. Semazzi, P. K. Snyder, and V. Kumar, “Data Guided Discovery of Dynamic Climate Dipoles.,” presented at the CIDU, 2011, pp. 30–44.